Low pass filter semiconductor structures for use in transducers for measuring low dynamic pressures in the presence of high static pressures

ABSTRACT

A semiconductor filter is provided to operate in conjunction with a differential pressure transducer. The filter receives a high and very low frequency static pressure attendant with a high frequency low dynamic pressure at one end, the filter operates to filter said high frequency dynamic pressure to provide only the static pressure at the other filter end. A differential transducer receives both dynamic and static pressure at one input port and receives said filtered static pressure at the other port where said transducer provides an output solely indicative of dynamic pressure. The filter in one embodiment has a series of etched channels directed from an input end to an output end. The channels are etched pores of extremely small diameter and operate to attenuate or filter the dynamic pressure. In another embodiment, a spiral tubular groove is found between a silicon wafer and a glass cover wafer, an input port of the groove receives both the static and dynamic pressure with an output port of the groove providing only static pressure. The groove filters attenuate dynamic pressure to enable the differential transducer to provide an output only indicative of dynamic pressure by cancellation of the static pressure.

RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.11/100,652, entitled Low Pass Filter Semiconductor Structures For Use InTransducers For Measuring Low Dynamic Pressures In The Presence Of HighStatic Pressures, filed Apr. 7, 2005 which is a continuation-in-part ofco-pending U.S. patent application Ser. No. 10/830,796, entitledPressure Transducer for Measuring Low Dynamic Pressures in the Presenceof High Static Pressures, filed Apr. 23, 2004, the entire disclosure ofwhich is hereby incorporated by reference as being set forth in itsentirety herein.

FIELD OF THE INVENTION

The invention relates to pressure transducers for measuring low dynamicpressures in the presence of high static pressures, and moreparticularly to improved low pass filter structures employed with suchtransducers.

BACKGROUND

During the testing of jet engines and in many other environments, it isoften desirable to measure both the static pressure and the dynamicpressure. The static pressure, in most instances, is usually very highand the dynamic pressure is much lower. The dynamic pressure is alsoassociated with a distinct frequency which occurs at a relatively highrate, for example 5000 cycles/second or greater. In this manner, thedynamic pressure may typically be 20 times less than the staticpressure. Hence, to measure static pressure, one requires a transducerwith a relatively thick diaphragm so that it can stand the high staticpressure. On other hand, such thick diaphragms have a poor response tolow pressure. Therefore, to measure static pressure and dynamic pressureis extremely difficult unless one uses a thick diaphragm in conjunctionwith a thin diaphragm. However, if one uses a thin diaphragm, then thisdiaphragm will rupture upon application of the high static pressurewhich also contains the dynamic pressure. One can think of the dynamicpressure as a relatively high frequency fluctuation on top of arelatively high constant static pressure. Thus, as one can ascertain,using a thick diaphragm to measure dynamic and static pressure is not aviable solution.

U.S. Pat. No. 6,642,594 ('594 Patent) entitled, “Single Chip MultipleRange Pressure Transducer Device”, which issued on Nov. 4, 2003 to A. D.Kurtz, the inventor herein and is assigned to Kulite SemiconductorProducts, Inc., the assignee herein, discloses problems with transducersresponsive to large pressures utilized to measure low pressures. Thus,pressure transducer adapted to measure relatively large pressurestypically suffer relatively poor resolution or sensitivity whenmeasuring relatively low pressures. This is because, as a span of thesensor increases, the resolution or sensitivity of that sensor at thelow end of the span decreases. An example of various piezoresistivesensors are indicated in the aforementioned '594 patent whereindifferent transducers have thinned regions having the same thickness,but different planar dimensions. In this manner, the thinned regionswill deflect a different amount upon application of a common pressurethereto, whereby when excited each of the circuits provides an outputindicative of the common pressure of a different operating range.

As indicated above, during the testing of jet engines there is a veryhigh static pressure which, for example, may be 100 psi. Present withthe static pressure is a low dynamic pressure, which may exhibitfrequencies in the range of 5000 Hz and above. As indicated, using ahigh pressure sensor to measure the static pressure will yield anextremely poor response to the dynamic pressure because of the smallmagnitude of dynamic pressure which can be, for example, about 5 psi.Therefore, it is desirable to use a relatively rugged pressuretransducer having a thick diaphragm to measure static pressure and toutilize another transducer on the same chip having a thinned diaphragmto measure dynamic pressure. Because the thinned transducer is exposedto static pressure both on the top and bottom sides, the static pressurecancels out and does not, in any manner, cause the thinned diaphragm todeflect. As described herein, both static and a dynamic pressure may beapplied to the rear side of the diaphragm by a reference tube ofsubstantial length. This reference tube, as will be explained, is ahelical structure and has a low resonant frequency. In this manner, whena small dynamic pressure is applied because of the low internalfrequency of the tube, the sensor will respond to the static pressureonly. The thinned diaphragm should be stopped for pressures in excess of25 psi, or some higher number than the desired dynamic pressure. Thelong reference tube can be made by taking a tubular structure andwrapping it such that it looks like a coil or spring. One end would beinserted into the transducer and other other end would be exposed topressure. In this manner, one can implement a transducer forsimultaneously measuring a low dynamic pressure in the presence of ahigh static pressure. Alternative transducer structures and methods formeasuring low dynamic pressure in the presence of high static pressureare also desired.

SUMMARY

A semiconductor filter is provided to operate in conjunction with adifferential pressure transducer. The filter receives both a high andrelatively low frequency static pressure attendant with a high frequencylow dynamic pressure at one end, and operates to filter the highfrequency dynamic pressure to provide only the static pressure at theother filter end. A differential transducer receives both dynamic andstatic pressure at one input port and receives the filtered staticpressure at the other port where the transducer provides an outputsolely indicative of dynamic pressure. The filter in one embodiment hasa series of etched channels directed from an input end to an output end.The channels are etched pores of extremely small diameter and operate toattenuate or filter the dynamic pressure. In another embodiment, aspiral tubular groove is formed between a silicon wafer and a glasscover wafer. An input port of the groove receives both the static anddynamic pressure with an output port of the groove providing only staticpressure. The groove filters attenuate dynamic pressure to enable thedifferential transducer to provide an output only indicative of dynamicpressure by cancellation of the static pressure.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a partial cross sectional view of a pressure transducer formeasuring low dynamic pressures in the presence of high staticpressures.

FIG. 2 is a partial cross sectional view of a pressure transducer formeasuring low dynamic pressures in the presence of high static pressuresemploying a semiconductor attenuator according to an embodiment of thepresent invention.

FIG. 3 is a series of top views showing alternate pore configurationsuseful for the semiconductor attenuator of FIG. 2.

FIG. 4 is a partial cross sectional view of a pressure transducerresponding to dynamic and static pressures employing a semiconductorhelical structure operating as an attenuator.

FIG. 5A shows a top view of the semiconductor attenuator utilized inFIG. 4; and FIG. 5B shows a cross sectional view of the semiconductorattenuator utilized in FIG. 4 taken through line B-B of FIG. 5A.

DETAILED DESCRIPTION

Referring to FIG. 1, there is shown a pressure transducer whichbasically consists of two leadless peizoresistive sensors 20 and 21mounted on header pins in accordance with the methods disclosed inKulite Patent No. 5,955,771 entitled, “Sensors for Use in HighVibrational Applications and Methods for Fabricating the Same” whichissued on Sep. 21, 1999 to A. D. Kurtz et al, the inventor herein andassigned to Kulite Semiconductor Products, Inc., the assignee herein.This patent is incorporated herein by reference.

Shown in FIG. 1 are two separate transducers 20 and 21 which arefabricated by the same process as according to the teachings of theabove-noted co-pending application and patent. The difference betweenthe two transducer or sensor structures is that the sensor structure onthe left has a diaphragm 20 which is thicker than the diaphragm 21 ofthe sensor structure on the right. Both sensors receive on their topsurfaces a pressure indicative of the static pressure (P_(s)) plus thedynamic pressure (P_(d)) (P_(s)+P_(d)). As indicated, the staticpressure (P_(s)) may be of a relatively high value and, for example,could be 100 psi or more. The dynamic pressure (P_(d)) appears as aripple on top of the static pressure (P_(s)) and is characterized by arelatively high frequency on the order of magnitude of 5000 Hz and aboveand a low value of 5 psi or less. Both sensors receive the combinationof the static plus the dynamic pressure shown in FIG. 1. Sensor 20, asindicated, has a thicker diaphragm and responds mainly to the staticpressure to produce at the output pins (15, 16) associated therewith, avoltage proportional to the static pressure. This voltage would indicatea static pressure of 100 psi or greater, whatever the case may be.

While the output of transducer 20 is also responsive to the dynamicpressure (P_(d)), the dynamic pressure (P_(d)) is an extremely smallpercentage of the total static pressure (P_(s)) and may, as indicated,be on the order of 5 psi or less. The thin diaphragm associated with thetransducer 21 will respond only to the dynamic pressure (P_(d)), as willbe explained. As seen in the Figure, transducer 21 has the static plusthe dynamic pressure applied to the top surface and is indicated againby Ps+Pd. Coupled to the bottom surface of the diaphragm is a tube orreference tube of an exceedingly long length, designated by referencenumeral 18. The tube 18 is coupled to the bottom surface of thediaphragm. Essentially, the tube 18 receives at an inlet both the staticand dynamic pressure, which is Ps+Pd.

The tube, as shown, is in helical form. It is well known that theresonant frequency f of such a tube, as, for example, an organ pipe, isgiven by f=c/(4 l), where c is the speed of sound, and l is the lengthof the tube. For instance, in air, where the speed of sound isapproximately 1200 feet per second, a tube length of 2½ feet will give aresonant frequency of 120 Hz. Thus, tube 18 acts as a low pass filterand will only pass frequencies which are below 120 Hz. In this manner,the dynamic frequency, which is 5000 Hz or greater, will not passthrough the tube 18. Therefore, the underside of the diaphragmassociated with transducer 21 only receives the status pressure (P_(s)).The static pressure (P_(s)) subtracted from the static pressure plus thedynamic pressure (P_(s)+P_(d)) supplied to the top surface of thediaphragm such that the output of the differential unit 21 provides apressure equal to the differential pressure (P_(d)). As seen, there is astop member associated with diaphragm 21. The stop member 25 assuresthat the diaphragm 21 will not deflect in a downward direction forpressures in excess of 25 psi, or some number higher than the desireddynamic pressure. The reference tube is fabricated by taking a tubularstructure, which may be metal or some other material, and wrapping itsuch that it looks like a coil or a spring where one end is insertedinto the transducer, as shown, and the other end is exposed to thestatic and dynamic pressure. Reference is made to U.S. Pat. No.6,642,594 entitled, “Single Chip Multiple Range Pressure TransducerDevice” issued on Nov. 4, 2003 to A. D. Kurtz, the inventor herein andassigned to the assignee herein, the entire disclosure of which ishereby incorporated by reference herein as well.

Therefore, the diaphragm associated with sensor 20 is intended foraccurately measuring static pressure. The sensor unit 21 will measuredynamic pressure because of the differential operation of the sensor 21and because of the tube. These dynamic pressures have relatively highfrequencies measured primarily by the first assembly 21, with the secondassembly 20 measuring the steady state pressure, which is a largepressure. The fabrication of stops, such as 25 for transducers, is wellknown in the art. See, for example, U.S. Pat. No. 4,040,172 entitled,“Method of Manufacturing Integral Transducer Assemblies EmployingBuilt-In Pressure Limiting” issued on Aug. 9, 1997 to A. D. Kurtz et aland is assigned to the assignee herein. See also U.S. Pat. No. 4,063,209entitled, “Integral Transducer Assemblies Employing Built-In PressureLimiting” issued on Dec. 13, 1997 to A. D. Kurtz et al. and assigned tothe assignee herein. The entire disclosures of U.S. Pat. Nos. 4,040,172and 4,063,209 are also incorporated by reference herein.

See also U.S. Pat. No. 6,595,066 issued on Jul. 22, 2003 to A. D. Kurtzet al. and is assigned to the assignee herein and entitled, “StoppedLeadless Differential Sensor”. This patent describes a leadless devicewhich is similar to the devices utilized in FIG. 1 which has a stopapparatus associated therewith. The sensor depicted in the '066 patentalso operates as a differential sensor with a Wheatstone bridge sensorarray. The output provides a difference between a pressure applied tothe top side of the sensor with respect to the force applied to thebottom side of the sensor. This sensor acts as the sensor 21 associatedand seen in FIG. 1. U.S. Pat. No. 6,595,066 is incorporated herein.

See also U.S. Pat. No. 6,588,281 issued on Jul. 8, 2003 entitled,“Double Stop Structure for a Pressure Transducer” issued to A. D. Kurtzet al. and assigned to the assignee herein. That patent shows a stopdevice in both first and second directions. As one can ascertain fromFIG. 1, a stop 25 is only required in the down direction. This is so, asthe large pressure Ps+Pd, as applied to the top surface, could rupturethe thin diaphragm if the pressure applied to the bottom surfacemomentarily is interrupted. In this manner, the diaphragm of the sensor21 will impinge upon the stop 25 to prevent the fracture of thediaphragm. The interruption of the pressure applied to the bottomsurface of the diaphragm could occur during pressure build-up or whenthe pressure source is first turned on or off. U.S. Pat. No. 6,588,281is also incorporated herein.

Referring now to FIG. 2, there is shown a transducer structure in whichthe helical tube 18 of FIG. 1 is eliminated. As the tube or organ pipeillustrated in FIG. 1 may be expensive and/or difficult to fabricate andincorporate, the semiconductor structure of FIG. 2 operates to emulatethe characteristics of the helical tube, including for example,frequency response, without providing such a helical tube.

FIG. 2 utilizes the same reference numerals as FIG. 1 for correspondingparts. It is seen that there are again two sensor diaphragms 20 and 21,each having piezoresistors located thereon and each receiving a pressureat a top surface which is the static plus dynamic pressure designated asP_(s)+P_(d).

A reference tube 25 is shown which receives the pressure P_(s)+P_(d) atthe inlet. Disposed between the back surface of transducer 21 and theinput of the reference tube 25 is a silicon wafer 26.

The wafer 26 has a plurality of holes or through channels, each having adiameter of less than about 0.001 inches and formed by etching ormicromachining, for example. In this case, as indicated in FIG. 2, thesmall diameter holes or apertures serve to attenuate any high frequencycomponents of the pressure caused by viscosity of the gas flowingthrough the apertures. The pressure and attenuation provided isdetermined by the diameter of the holes, the number of holes, as well asby the cavity volume on the underside of the sensing diaphragm.

The hole diameter, number of holes on the silicon wafer 26, and thecavity size are selected such that a desired filtering frequency can beobtained utilizing the formula:

$\tau = \frac{32\; \gamma \; {vVL}}{{AD}^{2}c^{2}}$

where τ=attenuation and

-   -   γ represents the ratio of specific heats; for air γ=1.4;    -   v represents the kinematic viscosity; for air v=14.5 m²/s or        0.0225 in²/s;    -   V represents the volume of the cavity or wafer;    -   L represents length of the pipe    -   A represents the total area of feeding pipes or apertures;    -   D represents the diameter of feeding pipe or apertures; and    -   c represents the speed of sound, which is about 1120 ft/s at        room temperatures.        For example; to achieve a cut-off frequency of 100 Hz, or a time        constant of 10 milli-seconds, the following parameters can be        selected.

-   D=0.0002 inch

-   A=0.000625 inch² assuming 25% pososity and a 0.050″ chip

-   c=1120 ft/s=13400 inch/2

-   L=0.005

-   V=0.001 in³

As one can see, attenuation is determined by the diameter of the hole aswell as the number of holes. The wafer having the silicon holes acts asa single hole of considerably longer length. The fabrication of holes insilicon is well understood and can be accurately controlled. See forexample, an article entitled “Porous Silicon/A New Material for MEMs”published in the IEEE 1996 by V. Lehmann of Siemens Ag Munchen, Germany,which describes a technique for the formation of pores or holes insilicon with high aspect ratios utilizing electrochemical etching ofn-type silicon wafers in hydrofluoric acid.

The wafer as shown in FIG. 2 is an n-type silicon wafer. As the articleindicates, porous silicon has been used for many years and may be formedon a silicon substrate during anodization in a hydrofluoric acidelectrolyte. Pore formation is present for anodic densities below acritical current density. The pore geometry can be controlled, as canthe pore cross section. The pore cross section can vary between a circleand a forearm star depending on the formation conditions. Subsequent tothe electrochemical pore formation, the cross section of the pores canbe made more circular by oxidation steps or can be made more squareshaped by anisotropic chemical etching for example using aqueous HF.

Referring to FIG. 3, there is shown a series of pore cross sections, allof which shapes can be formed during the etching process, and whichshapes have been described in the above-identified article. Whilecircular shapes may be preferred, the pores can be of a square or anyother suitable configuration represented by shapes designated A throughE in FIG. 3.

While the above-identified article describes process steps which arestandard techniques in microelectronic manufacturing, such techniquesmay be used to develop pore configurations in a silicon substrate whichenable communication between the bottom surface of the substrate to thetop surface of the substrate.

Based on the diameter of the pores and based on the width of the siliconwafer, one can therefore obtain the same frequency characteristics asare available by a helical tube. The bottom surface of wafer diaphragm21 thus receives only the static pressure (P_(s)), whereby the higherfrequency dynamic pressure is completely suppressed by the semiconductorwafer 26 having pores of configurations A-E as shown in FIG. 3.

The pore configurations A to E have various cross sections and willextend from the top surface of the wafer to the bottom surface of thewafer.

Referring now to FIG. 4 there is shown an alternate embodiment of asemiconductor arrangement that emulates the helical tube illustrated inFIG. 1 and that functions in a manner similar to the apertured waferstructure 26 illustrated in FIG. 2. FIG. 4 illustrates transducerdiaphragms 20 and 21 arranged in a housing whereby the static plusdynamic pressure (P_(s)+P_(d)) is applied to the top surface. Thereference tube 25 again receives the static and dynamic pressure, whichnow is applied to the bottom surface of a semiconductor wafer 28. Thesemiconductor wafer 28 has an input aperture 30 which is directed into acoiled hollow helical structure fabricated on the surface of thesemiconductor substrate. The helical structure has an output aperture 31communicating with the underside of the diaphragm 21.

Thus, the underside of the diaphragm 21 receives the static and dynamicpressure and because of the helical structure fabricated on thesemiconductor wafer 28, the dynamic pressure frequencies are againsuppressed.

Referring to FIG. 5, there is shown a top view of the wafer 28. Thewafer 28 as shown in a cross sectional view of FIG. 5 b has a bottomsilicon wafer 33 with a glass cover wafer 32. The silicon wafer is firstprocessed to provide a helical structure 40 on a top surface. Thehelical structure 40 is fabricated at a given depth. The helicalstructure communicates with an aperture 30 as shown in FIG. 4 to enablethe static plus the dynamic pressure to be applied to aperture 30.

The pressure is then circulated within the helical structure as coveredby the glass cover member 32 and at the output aperture 31 the pressureP_(s) which is the static pressure now applied to the underside of thediaphragm.

It is understood that there are numerous ways of fabricating helicalstructures in semiconductor material. These can be fabricated byutilizing stacking layers whereby a spiral coil is fabricated betweenthese layers and effectively constitutes a helical semiconductorstructure which manifests itself in having the same diameter and lengthas the helical tube shown in FIG. 1. In this case, the length of thespiral determines the frequency of operation according to the followingformula

$f = {\frac{c}{4L}\sqrt{1 - \left( \frac{\frac{4v}{D^{2}}}{\frac{c}{4L}} \right)^{2}}}$

where c=speed of sound, about 1120 ft/s

-   -   L=length of pipe or spiral;    -   D=diameter of pipe or spiral;    -   v=kinematic viscosity: for air, v is about 14.5 m²/s or 0.0225        in²/s.        In an exemplary configuration, a spiral of 0.005 inch diameter        and 1 inch in length achieves a filtering frequency f of 100 Hz.

By varying lengths and diameters of the holes, as for example concerningthe embodiment depicted in FIG. 2, one can tailor the frequencyattenuation to desired value. According to an aspect of the presentinvention, such frequency attenuation can be attained in an exceedinglysmall space. The structures as described herein can be mounted directlybehind a deflecting diagram and beyond the header, as shown in FIGS. 2and FIG. 4, for example.

It should be noted that after using standard micromachining techniques,a significant number of these structures could be made simultaneouslywithin a relatively small size (of silicon, for example). The processingtechniques, as indicated above, will enable such structures to beproduced, and hence produce reliable semiconductor attenuators orsemiconductor filters for use in static and dynamic pressuremeasurements.

While it is understood that the figures and descriptions hereinillustrate a dual transducer structure, it is understood that a singletransducer can be utilized, whereby static and dynamic pressure appliedto one surface and static and dynamic pressure are applied to the bottomsurface or the opposing surface via a semiconductor attenuator such as asemiconductor wafer having through pores from the top to the bottomsurface. Alternately, a semiconductor helical arrangement having ahollow passageway from an input port which receives the static anddynamic pressure to an output port which will only allow the staticpressure to pass due to the length and diameter of the helix can beemployed.

It is therefore understood that the above-noted semiconductor structuresmay replace the mechanical helical design in a more efficient andcompact structure while enabling a great number of applications to beprovided. While the above noted exemplary embodiments are preferred, itis also understood that alternative embodiments can be employedaccording to the teachings of this invention.

For example, a single transducer such as that depicted by referencenumeral 21 can be utilized to produce an output indicative of dynamicpressure and whereby the static pressure would be cancelled. These andalternate embodiments can be ascertained by one skilled in the art andare deemed to be encompassed with the spirit and scope of the claimsappended hereto.

1-21. (canceled)
 22. A differential pressure transducer with a filterfor suppressing a low dynamic pressure P_(d) in presence of a highstatic pressure P_(s), said differential pressure transducer comprising:a semiconductor substrate having a top and a bottom surface; a firstpiezoresistive sensor positioned on said top surface on a firstdiaphragm; a second piezoresitive sensor positioned on said top surfaceon a second diaphragm and adjacent to said first sensor; wherein saidsemiconductor substrate with first and second piezoresistive sensors isconfigured to act as a differential pressure transducer; and a filtercomprising a semiconductor wafer positioned proximate said bottomsurface of said semiconductor substrate and having high frequencyattenuation means positioned thereon to enable said semiconductor waferto receive both said dynamic and static pressure at an input end and toprovide only said static pressure at an output end proximate to saidbottom surface of said semiconductor substrate to provide an outputstrictly proportional to said dynamic pressure, whereby when saiddifferential transducer receives said low dynamic pressure and said highstatic pressure on said top surface of said substrate and receives onlysaid high static pressure on said bottom surface through said filteroutput, said differential transducer provides an output proportional toonly said dynamic pressure due to cancellation of said high staticpressure.
 23. The differential pressure transducer of claim 22, whereinsaid semiconductor substrate is silicon.
 24. The differential pressuretransducer of claim 22, wherein said semiconductor wafer has a series ofetched apertures extending from an input end of said wafer, saidapertures being of a number and diameter to attenuate said dynamicpressure at said output to cause only said static pressure to appear atsaid output.
 25. The differential pressure transducer of claim 22,wherein said semiconductor wafer of said filter is silicon.
 26. Thedifferential pressure transducer of claim 25, wherein said silicon isn-type silicon.
 27. The differential pressure transducer of claim 22,wherein said filter has a filtering frequency T is obtained according tothe formula: $\tau = {\frac{32\gamma \; {vVL}}{{AD}^{2}c^{2}}.}$ γis the ratio of specific heats, v is the kinematic viscosity, V is thevolume of the wafer, A is the total area of apertures, D is the diameterof the apertures, L represents length of the tube, c is the speed of thesound.
 28. The differential pressure transducer of claim 22, whereinsaid semiconductor wafer has a spiral pattern formed on a top surface,said spiral formed to a given depth with said spiral being of a givenlength, with a first end of said spiral coupled to an input apertureextending from an input port of said spiral to a bottom input end ofsaid wafer to enable said spiral to receive both said static and dynamicpressure, further comprising: a cover member secured to said top surfaceof said wafer to cover said spiral to form a spiral tube, an output portaperture in said cover member for communicating with said other end ofsaid spiral, with said cover member aperture proximate to saidsemiconductor substrate to provide at said output port said staticpressure.
 29. The differential pressure transducer of claim 28, whereinsaid cover member is fabricated from glass.
 30. The differentialpressure transducer of claim 28, wherein the length and diameter of saidspiral tube is selected according to the formula:$f = {\frac{c}{4L}\sqrt{1 - \left( \frac{\frac{4v}{D^{2}}}{\frac{c}{4L}} \right)^{2}}}$where, f is the filtered frequency, d is the diameter of the tube, c isthe speed of the sound, v is the kinematic viscosity, and L is thelength of the tube.
 31. The differential pressure transducer of claim22, wherein said second diaphragm is thicker than said first diaphragm.